Theorem bd0 15578

Description: A formula equivalent to a bounded one is bounded. See also bd0r 15579. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 15567	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15566

This theorem was proved from axioms:  ax-mp 5  ax-bd0 15567

This theorem is referenced by:  bd0r  15579  bdth  15585  bdnth  15588  bdnthALT  15589  bdph  15604  bdsbc  15612  bdsnss  15627  bdcint  15631  bdeqsuc  15635  bdcriota  15637  bj-axun2  15669